Tag: statistics and probability

Questions Related to statistics and probability

Coefficient of range $5, 2, 3, 4, 6, 8, 10$ is?

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $\dfrac{2}{3}$

  2. $\dfrac{1}{3}$

  3. $\dfrac{3}{5}$

  4. $\dfrac{1}{2}$

Show answer
Correct Option: A
Explanation:
${ x _{ m } }=10{ x _{ 0 } }=2$
coefficient of range 
$\begin{array}{l} =\frac { { { x _{ m } }-{ x _{ 0 } } } }{ { { x _{ m } }t{ x _{ 0 } } } }  \\ =\frac { { 10-2 } }{ { 10+2 } } =\frac { 8 }{ { 12 } } =\frac { 2 }{ 3 }  \end{array}$

The highest score of a certain data exceeds in lowest score by $16$ and coefficient of range is $\cfrac{1}{3}$. The sum of the highest score and the lowest score is

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $36$

  2. $48$

  3. $24$

  4. $18$

Show answer
Correct Option: B
Explanation:

Let the highest score be $x _{m}$ and 

the lowest score be $x _{0}$
Given that highest score exceeds lowest score by $16$
$\implies x _m=x _0+16\implies x _m-x _0=16$ ————(1)

Coefficient of range is given by $\dfrac{x _m-x _0}{x _m+x _0}$

Given that coefficient of range is $\dfrac 13$

$\implies \dfrac{x _m-x _0}{x _m+x _0}=\dfrac 13$ ———(2)

Substitute (1) in (2) we get

$\dfrac{16}{x _m+x _0}=\dfrac 13$

$\implies x _m+x _0=16\times3=48$

Therefore sum of the highest score and lowest score is $48$

For a frequency distribution $8^{th}$ decile is computed by the formula

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $ \displaystyle D _{8}= l _{i}+\frac{\frac{N}{8}-C}{f}\times \left ( l _{2}-l _{1} \right )$

  2. $ \displaystyle l _{1}+\frac{\frac{8N}{10}-C}{f}\times \left ( l _{2}-l _{1} \right )$

  3. $ \displaystyle D _{8}= l _{1}+\frac{\frac{N}{10}-C}{f }\times \left ( l _{2}-l _{1} \right )$

  4. $ \displaystyle l _{1}+\frac{\frac{10N}{8}-C}{f }\left ( l _{2}-l _{1} \right )$

Show answer
Correct Option: B
Explanation:

A decile is any of the nine values that divide the sorted data into ten equal parts, so that each part represents 1/10 of the sample or population.
For a continuous distribution, the formula for $r^{th}$ decile is given by $D _r = l _1 + \frac{\frac{rN}{10} - C}{f} \times (l _2 - l _1)$
Substituting r = 8, we have $D _8 = l _1 + \frac{\frac{8N}{10} - C}{f} \times (l _2 - l _1)$ 

If $n> 1, x> -1, x\neq 0$, then the statement $\left ( 1+x \right )^{n}> 1+nx$ is true for

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $ \;n\;\epsilon \;N$

  2. $\forall \;n\;> 1$

  3. $x> -1 \;and\; x\neq 0$

  4. None of these

Show answer
Correct Option: A
Explanation:

$P(1)$ is not true 


For $n=2,P\left( 2 \right) :{ \left( 1+x \right)  }^{ 2 }>1+2x$ is true if $x\neq 0$

Let $P(k):{ \left( 1+x \right)  }^{ k }>1+kx$ be two 

$\therefore{ \left( 1+x \right)  }^{ k+1 }=\left( 1+x \right) { \left( 1+x \right)  }^{ k}>\left( 1+x \right) \left( 1+kx \right)> 1+\left( k+1 \right) x+k{ x }^{ 2}>1+\left(k+1\right) x$

$\left( \because k{ x }^{ 2 }>0 \right) $
$\therefore$ By PMI
Given statement is true for every $n\in N$.

The coefficient of mean deviation from median of observations  $40, 62, 54, 90, 68, 76$  is

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $2.16$

  2. $0.2$

  3. $5$

  4. None of these

Show answer
Correct Option: B
Explanation:

Arrange the given observations in ascending order
$40,54,62,68,76,90$
Here, number of terms $n=6 (even) $
$\displaystyle \therefore $ Median (M) $\displaystyle =\frac{\left ( \frac{n}{2} \right )th:term+\left ( \frac{n}{2}+1 \right )th:term}{2}=\frac{62+68}{2}=65$

$\Sigma \left | x _{i}-M \right |=25+11+3+3+11+25=78$
Mean deviation from median $\displaystyle =\frac{\Sigma \left | x _{i}-M \right |}{n}=\frac{78}{6}=13 $
$\therefore $ Coefficient of M.D.=$\displaystyle =\frac{M.D.}{median}=\frac{13}{65}=0.2$

The coefficient of mean deviation from median of observations 40, 62, 54, 90, 68, 76 is

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. 2.16

  2. 1.2

  3. 5

  4. none of these

Show answer
Correct Option: B
Explanation:

Arranging the given data in ascending order
40,54,62,68,76,90
Here, $n=6 (even)$
$M= \dfrac{\text{value of }3^{rd}\text{observation}+\text{value of }4^{th}\text{observation}}{2}$
Median $M=\dfrac{62+68}{2}=65$

Mean deviation about median $M.D=\dfrac{|40-65|+|54-65|+|62-65|+|68-65|+|76-65|+|90-65|}{65}$

$=\dfrac{25+11+3+3+11+25}{65}=1.2$

The difference between the maximum and the minimum observations in the data is

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. class interval

  2. frequency

  3. cumulative frequency

  4. range

Show answer
Correct Option: D
Explanation:

The difference between maximum and the minimum observation in the data is range.

For example, suppose an experiment involves finding out the weight of lab rats and the values in grams are 320, 367, 423, 471 and 480. In this case, the range is simply computed as 480-320 = 160 grams.

The coefficient of range of a set of data is given to be $\dfrac18$. Then the ratio of the maximum value in the data to the minimum value is:

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $\dfrac81$

  2. $\dfrac98$

  3. $\dfrac97$

  4. $\dfrac87$

Show answer
Correct Option: C
Explanation:

Coefficient of range of a set of data is given by $\dfrac{max-min}{max+min}$
$\dfrac{max-min}{max+min}=\dfrac{1}{8}$
$8max-8min=max+min$
$7max=9min$
$\dfrac{max}{min}=\dfrac{9}{7}$

The following are the wages of 8 workers in a factory. Find the range and coefficient of range. Wages are in dollars: 1400, 1450, 1520, 1380, 1485, 1495, 1575, 1440.

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $0.0231$

  2. $0.03112$

  3. $0.66$

  4. $0.02314$

Show answer
Correct Option: C
Explanation:

The largest value of data is $x _m=1575$

The smallest value of data is $x _0=1380$
Range$=x _m-x _0=1575-1380=195$

Coefficient of data$=\dfrac{1575-1380}{1575+1380}=\dfrac{195}{2955}=0.0659\approx 0.66$

If the coefficient of range is $0.18$ and the largest value is $7.44$,then the smallest value is?

range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

  1. $3.23$

  2. $4.15$

  3. $5.17$

  4. $5.14$

Show answer
Correct Option: C
Explanation:

Coefficient of range$=\dfrac{x _m-x _0}{x _m+x _0}=\dfrac{7.44-x _0}{7.44+x _0}$

$0.18(7.44+x _0)=7.44-x _0$
$1.18x _0=7.44-7.44\times 0.18$
$1.18x _0=6.1008$
$x _0=5.17016\approx 5.17$